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Double polarisation in nonlinear vibrating piano strings
Jin-Jack Tan, Cyril Touzé, Benjamin Cotté
To cite this version:
Jin-Jack Tan, Cyril Touzé, Benjamin Cotté. Double polarisation in nonlinear vibrating piano strings.
Vienna talk 2015 on music Acoustics, Sep 2015, Vienna, Austria. �hal-01194580�
DOUBLE POLARISATION IN NONLINEAR VIBRATING PIANO STRINGS
Jin Jack Tan, Cyril Touz´e, Benjamin Cott´e
IMSIA,
Institute of Mechanical Sciences and Industrial Applications,
ENSTA-ParisTech - CNRS - EDF - CEA,
Universit´e Paris-Saclay,
828 Boulevard des Marchaux, 91762 Palaiseau Cedex, France
jtan@ensta.fr
ABSTRACT
The present work studies the double polarisation phenomenon observed in vibrating piano strings. From the experimental view-point, it is known that when a string is given an initial displace-ment in one transverse direction (e.g. hammer excitation in the vertical plane), the second transverse displacement (e.g. in the horizontal plane) is also excited after a few milliseconds and the amplitude can be of similar order to the first transverse dis-placement. This phenomenon contributes to a characteristic pi-ano sound feature called the ”double decay”. The purpose of this study is to investigate the role of nonlinearities in inducing double polarisations. The nonlinear vibrations of the strings are studied with a two-degrees-of-freedom (dofs) system extracted from the Kirchhoff-Carrier string equations. The method of multiple scales is used to study the free vibrations of two po-larisations having nearly equal eigenfrequencies and thus pre-senting a 1:1 internal resonance. For an imperfect string with slightly different eigenfrequencies between the two polarisa-tions, it is found out that depending on the energy of the excita-tion, an uncoupled transverse mode can develop into a coupled mode where there is energy exchange between the two trans-verse polarisations. The coupled mode is stable and the string oscillates in an elliptic path. Numerical experiments are also carried out, confirming the findings of the analytical approach.
1. INTRODUCTION
This study on the double polarisation of piano strings is con-tained in a larger project which aims at developing a physically-based time-domain piano sound synthesis model. The work was pioneered by Chabassier [1] who proposed a refined complete model of a grand piano [2, 3]. However a few points still need additional developments. For example, one tonal feature that was not observed in the simulated sound is the double decay, i.e. where the temporal signal exhibits two envelopes with dif-ferent decay rates. It is understood that one of the contribut-ing factors of double decay is the double polarisation of piano strings, as first reported by Weinreich [4]. The double polari-sation phenomenon is not modeled in Chabassier’s work which may explain the absence of double decay in her modeled pi-ano sound. The double polarisation can be caused by various factors, such as nonlinearities of piano strings, asymmetry and complex boundary conditions, coupling to unison strings etc. [5, 6, 7, 8, 9].
In this contribution, we focus on the nonlinearity experi-enced by piano string vibrating at large amplitudes as a possi-ble cause for explaining the birth of doupossi-ble polarisation. More particularly the two fundamental eigenfrequencies of a string
with double polarisation are known to have close values, thus showing a 1:1 internal resonance. It is also known from other studies on nonlinear vibrations [10, 11] that thanks to a 1:1 in-ternal resonance, energy can be exchanged between vibration modes so that even if the motion is initiated along one polari-sation only, the nonlinearity can make this motion unstable, so that eventually a coupled vibration arises with the two polari-sations involved. The objective of the present contribution is thus to clearly establish if the nonlinearity can be the cause of this coupling, as well as to highlight the main parameters gov-erning the transfer of energy. The 1:1 internal resonance has already been studied in the case of forced vibrations, see e.g. [12, 10, 11]. Here our interest is in the case of free vibration for which only Manevitch and Manevitch present a detailed investi-gation [13]. The complete problem will hence be fully revisited and applied to the specific case of strings.
The article is organised as follows. First, a system of non-linear Kirchhoff-Carrier string equations is presented in sec-tion 2 and solved via the multiple scales method in secsec-tion 3. The results are presented and discussed in section 4. The pa-per continues with a numerical expa-periment to demonstrate the double polarisations in section 5 before wrapping up with con-clusion in section 6.
2. KIRCHHOFF-CARRIER EQUATION
The Kirchhoff-Carrier equations for a freely vibrating fixed-fixed string with two polarised displacements u1 and u2 read
[14, 15, 11]:
ρA ¨u1− (T0+ N ) u001 = 0, (1a)
ρA ¨u2− (T0+ N ) u002 = 0, (1b)
where N is the axial tension created by the large amplitude mo-tions and the coupling with the transverse motion. It reads:
N =EA 2L ZL 0 u01 2 + u02 2 dx. (2)
In this set of equations, L is the length, A the crosse section area, E Young’s modulus, T0 the tension and ρ the density.
Among others, underlying assumptions are that the inertia of the longitudinal component is negligible, see e.g. [16, 14] for more details. Following [11], the equations can be made nondi-mensional for a more general treatment.
ex-pressed in modal form: u1(x, t) = K X k=1 Φk(x)pk(t), u2(x, t) = K X k=1 Φk(x)qk(t), (3) where Φk(x) are the mode shapes, and {pk(t), qk(t)}k≥1the
modal coordinates. Restraining only the fundamental mode in the truncation for each polarisation (i.e. K = 1) and using the usual projection (Galerkin) technique, on obtains the following two dofs nonlinear system for the two modal coordinates p, q related to each polarisation:
¨
p + ω12p + εΓ1p3+ C1pq2 = 0, (4a)
¨
q + ω22q + εΓ2q3+ C2qp2 = 0, (4b)
where ε is a small parameter arising from the nondimensionali-sation [11], ε = EAd2/T0L2, with d the diameter of the string.
For a perfect string, the two eigenfrequencies are equal so that ω2 = ω1 = π [11]. The same holds for the nonlinear
coeffi-cients, in a perfect case we have Γ1= Γ2= C1= C2= π4/2.
However, acknowledging that such perfect string does not exist in reality, a detuning parameter σ1is introduced so that the two
eigenfrequencies of the two modes are possibly slightly differ-ent,
ω2= ω1+ εσ1. (5)
Following the same lines, the system will be studied with gen-eral and different nonlinear coefficients so as to draw the com-plete picture for the nonlinear string. In a given experimental case, a procedure would be needed for identifying these coeffi-cients.
3. MULTIPLE SCALES METHOD
The system in equation (4) is solved by the multiple scales method which describes the original system to be function of multiple independent time scales. Introducing the ”fast” and ”slow” time scales:
T0= t, (6a)
T1= εt, (6b)
p and q can take the following form,
p(t) = p0(T0, T1) + εp1(T0, T1) + O(ε2), (7a)
q(t) = q0(T0, T1) + εq1(T0, T1) + O(ε2), (7b)
where p0and q0can be written as:
p0= A(T1) exp(iω1T0) + c.c, (8a)
q0= B(T1) exp(iω2T0) + c.c, (8b)
where c.c stands for complex conjugates. A and B are unknown complex functions of T1. Substituting equation (7) into (4) and
grouping all the resonant terms for 1:1 resonance up to order ε, one obtains the solvability conditions. By writing A and B in polar form:
A(T1) = a(T1) exp(iα(T1)), (9a)
B(T1) = b(T1) exp(iβ(T1)), (9b)
the solvability conditions can be broken down into a set of four dynamical equations (two for amplitude a and b, two for the
phases α and β): a0= −C1 2ω1 ab2sin(γ2− γ1), (10a) γ10 = 3Γ1 ω1 a2+C1 ω1 b2[2 + cos(γ2− γ1)] , (10b) b0= C2 2ω2 ba2sin(γ2− γ1), (10c) γ20 = 3Γ2 ω2 b2+C2 ω2 a2[2 + cos(γ2− γ1)] + 2σ1, (10d) where γ1= 2α, γ2= 2β + 2σ1T1. (11)
The introduction of equation (11) is necessary so that the sys-tem is made autonomous (not directly dependent on any time scales).
3.1. Uncoupled solutions
Let us first consider the uncoupled solutions. They correspond to the motions of the strings that are either in the horizontal direction, or in the vertical one. The first set of uncoupled solu-tions is found by setting b = 0, in Eqs. (10). The 4-dofs system then degenerates into a two dofs and reads:
a0= 0, (12a) γ01=
3Γ1
ω1
a2, (12b)
The equations can be easily integrated to give:
a = Ca, (13a)
α = 3Γ1 2ω1
a2T1+ αa, (13b)
where Caand αaare both integration constants independent of
T1. Using this result, the solution p0can be expressed as:
p0= 2a cos [ωN Lt + αa] , (14) where ωN L= ω1 1 + ε3Γ1 2ω2 1 a2 . (15) The first order solution is thus a periodic orbit where only p is involved in the vibration since setting b = 0 implies q = 0. The nonlinear frequency of oscillation ωN Ldepends on the
ampli-tude a, a usual feature in nonlinear oscillations. For a positive value of Γ1, which is the case for strings, the nonlinearity is of
the hardening type, i.e. the oscillation frequency increases with the amplitude.
A similar exercise can be done for the other uncoupled case, i.e. a = 0 and one would obtain a similar result for q0,
q0= 2b cos [ωN Lt + βb] (16) where ωN L= ω2 1 + ε3Γ2 2ω2 2 b2 (17) One can notice the similarity between the two solutions, coming from the fact that the uncoupled solutions are ruled out by clas-sical Duffing equations. The nonlinearity is completely gov-erned by coefficients Γ1 and Γ2. Figure 1 shows the
back-bone curve (amplitude-frequency relationship, Eqs. (15)-(17)) for ε = 0.163, a typical value for a string that has been com-puted using standard values from [1]. The value Γ = 0 is used
3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 frequency, ωNL amplitude, a Γ 1=0 Γ1=π4/4 Γ1=π4/2
Figure 1: Relationship between a and ωN L for various Γ1 at
ω1= π, ε = 0.163.
as an eyeguide to recall that for linear vibrations, the oscilla-tion frequency is independent from the amplitude. The values Γ = π4/2, ω = π are the standard values for perfect strings. Finally Γ = π4/4 is plotted as an intermediate case between the linear string and the perfect nonlinear string. The figure allows one to estimate the deviation (in radian frequency) of the oscil-lations with respect to the linear eigenfrequency, as a function of the vibration amplitude.
3.2. Coupled solutions
Let us now investigate the coupled solutions of the conservative system given by Eqs. (10). Stationary oscillations occurs at a given energy level so that coupled solutions can be searched for fixed amplitudes, by imposing a0 = b0 = 0. This is also in the line of the uncoupled cases where periodic solutions were found for fixed amplitudes and only phase variations, from which the nonlinear amplitude-frequency relationship were derived. From equation (10a) and (10c), it is obvious that for coupled solu-tions to exist (i.e. a 6= 0, b 6= 0), one must have mandatory sin(γ2−γ1) = 0. This implies in particular that cos(γ2−γ1) =
±1. Interestingly, for each case of the possible value of the co-sine, the two polarisations p and q are related in a different man-ner. For cos(γ2− γ1) = +1, a simple algebra on the system
shows that the solutions p and q are related by the following relationship:
q p = ±
b
a, (18) while for cos(γ2− γ1) = −1, p and q are related by:
q2
4b2 +
p2
4a2 = 1. (19)
These particular forms expressed by the coupled solutions has already been commented by Manevicth and Manevitch [13], who refers to them respectively as ”normal mode” (NM) (for-mer case, Eq. (18)), and ”elliptic mode” (EM) (latter case, and Eq. (19)). This peculiar relationship expressed for each time between the solution amplitudes leads to a particular motion which is sketched in Figure 2 in the (p, q) (displacements) plane. The elliptic mode appears particularly interesting for us as it corresponds to the whirling motion observed in piano strings [9].
The nonlinear amplitude-frequency relationships defining the backbone curves for coupled solutions can be found out
q
p p
q
(a) normal mode (b) elliptic mode
Figure 2: Illustrations of the two modes of coupled solutions. Image not to scale.
by setting cos(γ2− γ1) = ±1 in Eqs. (10b)-(10d), where the
right-hand sides become constants, so that one can retrieve α, β and subsequently the nonlinear frequency ωN Lfor the coupled
modes as: ωN L= ω1 1 + ε 3Γ1 2ω2 1 a2+rC1 2ω2 1 b2 . (20)
In this equation (and in the remainder of the article), the pa-rameter r is such that r = 3 for normal mode, and r = 1 for elliptic modes respectively. As can be seen in equation (20), the nonlinear frequency is influenced by the amplitude of both polarisations a and b. The equation forms a surface on the 3D space defined by (a, b, ωN L). However curved line
solu-tions are awaited instead of a whole surface family. Noting that γ2− γ1 = nπ from the necessary conditions of existence for
coupled solutions, one has thus γ20 − γ 0
1 = 0. Using such
rela-tion, one can obtain: rC1 ω1 −3Γ2 ω2 b2+ 3Γ1 ω1 −rC2 ω2 a2 = 2σ1. (21)
This relationship expresses the link between the amplitudes a and b for both normal (r = 3) and elliptic (r = 1) coupled modes. The conjuction of Eqs (20) and (21) define the backbone curves for the coupled modes.
Let us now investigate how the coupled solutions can be related to the uncoupled ones. By setting a = 0 or b = 0 in Eq. (21), it can be seen that either of the uncoupled modes can branch into the coupled modes provided the following condi-tions are met:
For uncoupled modea 6= 0
a2≥ 3Γ 2σ1 1 ω1 − rC2 ω2 , (22)
For uncoupled modeb 6= 0
b2≥ rC 2σ1 1 ω1 − 3Γ2 ω2 , (23)
These equations provide a limit value, in terms of amplitudes of the uncoupled modes, for which the coupled solutions can develop. Below these limit value, only uncoupled solutions exist. It must be noted that in certain cases, the RHS of both the equations can have two positive values of which the lower one indicates the bifurcation point where the uncoupled solu-tion branches into the coupled solusolu-tion while the higher one in-dicates the point where the coupled solution collapse and enter the uncoupled solution [13].
3.3. Stability analysis
To obtain the stability of the coupled modes, Eqs (10) can be reduced to a 3-DOF system by taking the difference between the two phases or even further to a 2-DOF system as demon-strated by Manevitch and Manevitch [13]. The advantage of using such an approach is that the coupled solutions are then real fixed points of the 3- or 2-dofs systems, so that the usual tools from dynamical system theory can be used for investigat-ing stability. One can thus construct the Jacobian matrix of the corresponding system and solve for the eigenvalues. Using ei-ther system will result in the same following stability criteria for the coupled modes:
ω2Γ1
C2ω1
+Γ2ω1 C1ω2
< 2 for normal modes, (24a) ω2Γ1
C2ω1
+Γ2ω1 C1ω2
>2
3 for elliptics modes. (24b) It is interesting to see that the stability of the coupled solutions does not depend on the energy but rather on the physical param-eters of the system (i.e. eigenfrequencies and nonlinear con-stants). It also means that regardless of level of excitation, ex-hibition of stable normal or elliptic modes are pre-determined.
To conclude the analytical study, let us investigate the sta-bility of uncoupled solutions and demonstrate how uncoupled solutions can become unstable in favour of a coupled one. As noted by Manevitch and Manevitch [13], the stability of the un-coupled solutions is determined by the energy of the system. Using the same approach as for the coupled case does unfor-tunately not give a useful criteria for the stability of uncoupled solutions, which are found to be always unstable. Furthermore, Manevitch and Manevitch do not give an explicit proof of the stability of uncoupled solutions in [13]. The underlying prob-lem is that when setting either b = 0 or a = 0 in the system, the degeneracy is ill-conditioned so that the phase space shrinks down to a two dofs system where the perturbation brought by the other oscillators are not defined and thus cannot be studied. The solution is found from the forced and damped vibration cases by canceling the damping terms and identifying the exter-nal excitation frequency to the nonlinear oscillation frequency ωN L. Using the existence conditions derived in [10] from a
ge-ometric analysis in phase space, one can obtain the following instability regionsfor the uncoupled solutions:
For uncoupled modea 6= 0
ωN L= ω2+ εr
C2
2ω2
a2 (25)
For uncoupled modeb 6= 0 ωN L= ω1+ εr
C1
2ω1
b2 (26)
where r = 1 and r = 3 define the lower and upper bound of the instability region, that are simply related to the connection with either elliptic mode or normal mode. The instability affecting the uncoupled mode a 6= 0 originates from the eigenfrequency of the other uncoupled mode, ω2and vice versa. This is because
the existence of another uncoupled solution upsets the stability of the original uncoupled solution. It is also interesting to note that the point where the uncoupled solution changes its stability (either in losing or restoring it) is also the point where the un-coupled solution branches into a un-coupled solution (or a un-coupled solution leaves and enters the uncoupled solution). Finally, the stability of the uncoupled solutions are seen to depend on the coupling nonlinear coefficients C1and C2only, as can be
ex-pected.
4. RESULTS AND DISCUSSIONS
3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 ω NL a, b
Figure 3: Amplitude-frequency relationships for the strings, perfect case with ω1 = ω2 = π (no detuning: σ1 = 0), and
equal nonlinear coefficients C1 = C2 = Γ1 = Γ2 = π4/2.
The black and blue lines (overlapped with each other) in-dicate the two uncoupled modes while the red line indicates the elliptic mode. ε = 0.163.
In this section, several case studies are made to demonstrate the properties of the system. Firstly, a perfect string case is con-sidered (i.e. σ1 = 0, C1 = C2 = Γ1 = Γ2). For this case,
the two amplitude-frequency relationships for uncoupled solu-tions given by Eqs. (15) and (17) are exactly the same so that the backbones in Fig. 3 collapse on the same curve (black and blue curves, only the blue being visible). For the coupled solutions, Eqs. (21) for the normal mode (r = 3) degenerates, indicating that no normal modes are possible in the perfect case. On the other hand, elliptic modes does however exist, and Eqs. (21) shows that they have same amplitude : a = b. Reporting in Eq. (20), one obtains the backbone curve for the coupled, el-lipitc modes in the perfect case as: ωN L = ω1(1 + 2εΓω1
1a
2).
This shows that coupled solutions have a stronger hardening be-haviour than uncoupled ondes, as reported in Fig. 3 with a red line. It must be noted that a 2D representation has been chosen for simplicity, by using the same axis for both amplitudes a and b, whereas the whole solutions should be plotted in a 3D space. In such 3D space (a, b, ωN L), uncoupled solutions are restricted
respectively to the planes (a, ωN L) and (b, ωN L), whereas the
coupled elliptic solutions is in the plane a = b.
Finally, examining the different stability conditions found in the previous section for both coupled and uncoupled modes leads to the conclusion that all the solutions reported in Fig. 3 are stable. This leads to the important conclusion that if the motion is excited on a given polarization, then it will stay on it for every time so that no whirling motion of the string would be observed. The coupled elliptic solutions could be observed only if very specific initial conditions are given to the string so that the motion is initiated along this mode.
Let us now turn to the more realistic case of an imperfect string. The most simple imperfection with a slight detuning between the two eigenfrequencies of the polarisation is investi-gated, by setting σ1= 1, and keeping all the nonlinear coupling
coefficients equal: C1= C2= Γ1= Γ2. The backbone curves
are represented in Fig. 4, where now the two uncoupled solu-tions (black and blue lines) are different and originates respec-tively from ω1and ω2= ω1+ εσ1. Eqs (21) shows once again
3 3.5 4 4.5 0 0.1 0.2 0.3 0.4 0.5 ω NL a, b ω 2 ω 1 stability curve
Figure 4: Amplitude-frequency relationships for periodic solu-tions of the nonlinear string, imperfect case with σ1 = 1, all
other coefficients being the same as in Fig. 3: C1 = C2 =
Γ1 = Γ2 = π4/2, ω1 = π, ε = 0.163. Black :
uncou-pled solution a 6= 0, blue uncoupled solution b 6= 0, red coupled elliptic mode. Dashed lines indicates instability. In brown is given the instability limit predicted by Eq. (25).
that in this case normal modes are not likely to exist. Only ellip-tic modes are possible. The instability condition for uncoupled solutions, provided by Eq. (25)-(26), shows that:
• The uncoupled solutions with b 6= 0 are always stable. • On the other hand, uncoupled solutions with a 6= 0 can
be unstable and branch on an elliptic mode, recovering also the criteria given by Eq. (22).
The instability region for uncoupled mode with a 6= 0 given by Eq. (25) has two curves. The one for normal mode (r = 3) is not relevant. Finally only the instability line with r = 1 is meaningful, and is represented as a brown line in Fig. 4. The crossing between the sdof uncoupled solution and the instabil-ity limit occurs exactly when condition (22) is fulfilled. From this point, uncoupled solutions are unstable, and the branch of elliptic mode solutions (red line) emerges. Once again a 2D representation has been chosen for simplicity, the reader must however keep in mind that the red line is neither in the plane (a, ωN L), nor in (b, ωN L), but really develop in the full 3D
space and is not contained within a plane. In particular the crossing between the coupled solution (red line) and uncoupled (b 6= 0, blue curve) is only a matter of the representation used but does not exist in the full 3D space.
The important conclusion that can be drawn from this study is that as soon as an imperfection is taken into account, an un-stable region in the backbone curve for uncoupled modes exist. Once the limit amplitude exceeded, uncoupled solutions are un-stable so that even though an initial condition is given for that polarisation, an energy transfer will occur so that eventually the system would settle on the stable elliptic mode. Interestingly, one can notice from Eq. (25) that the smallest the detuning σ1,
the smallest the amplitude limit for unstable solutions occurs. Hence in order to observe easily this phenomenon, the detuning need not be zero, but should be as small as possible.
5. NUMERICAL EXPERIMENTS
The main analytical findings of the previous section is now com-pared to direct numerical simulations of the original system
given by Eqs. (4). More particularly, the mots interesting case of the imperfect string is considered, with a slight detuning be-tween the two eigenfrequencies of the two polarisations. The equations of motion are directly integrated numerically in time with a fourth-order Runge-Kutta method. The experiment is carried out with varying level of initial excitation on the first mode only, mimicking the case of a piano string being struck with a hammer of increasing velocities, and hence exciting the string in only one polarization. The values of the case previ-ously studied in Fig. 4 are selected. The analytical study pre-dicts that the uncoupled solution should be stable as long as a < 0.25. A first case is thus studied below this limit value, with an initial condition in displacement only, with p(t = 0) = 0.4 and q(t = 0) = 1e − 4. Note that, from Eqs. (9)-(14), a factor 2 between a (resp. b) and the amplitude solution in time for p(t) (resp. q(t)), is present, so that the limit amplitude for stability for p(t = 0) is 0.5. The direct time integration is represented in Fig. 5. One can observe that below the amplitude where in-stability occurs, the second coordinate q(t) stays at negligible values around 1e − 4. 0 20 40 60 80 100 120 140 160 −0.5 0 0.5 time [nondim] p(t), q(t) 0 20 40 60 80 100 120 140 160 −0.5 0 0.5 time [nondim] p(t), q(t)
Figure 5: Time responses from direct numerical integration of Eqs. (4) and for varying levels of initial displacements p(t = 0). At the top, p(t = 0) = 0.4, below the instability limit; at the bottom, p(t = 0) = 0.8, above the instability limit. Black line correspond to p(t) and blue line to q(t).
For an initial condition above the instability limit, p(t = 0) = 0.8, one can observe that an energy exchange between the two modes occurs and q(t) reaches values up to 0.6. The energy then gets back and forth between the two oscillators, a typical feature of nonlinear conservative oscillations [12]. This numerical experiment confirms the instability of the uncoupled solution. It also shows that, for a given piano string, and for amplitude of excitation that are above a certain threshold that can be predicted, a motion initiated along a single polarisation can be transformed into a coupled whirling motion, as observed experimentally.
6. CONCLUSION
A detailed study has been conducted to examine the two polar-isations of nonlinear vibrating strings, and to show if the geo-metric nonlinearity due to large-amplitude vibration can be re-sponsible of the coupling between the two polarizations, even
though the motion is initiated along a single direction. An an-alytical study has been conducted using a two-modes approxi-mation for the Kirchhoff-Carrier equation, and then the method of multiple scales. The frequency-amplitude relationships of the two polarisations and the ways whirling motion can take place are identified. The main finding of the anlytical study re-veals that an imperfection is needed in order to make uncoupled solutions unstable. When all the coefficients are equal, in the mathematical case of a perfect string, then the periodic solu-tions are all stable. When a detuning is considered between the two eigenfrequencies, which is always the case in practice, then uncoupled solutions can become unstable and whirling motions of the string can take place even though the motion is initiated along one polarisation only. These results also clearly demon-strate that the geometric nonlinearity can be a potential cause for the whirling motions observed in real piano strings. Future work will consider experimental validations of these findings. The imperfections brought by the specific boundary conditions of the piano string will also be studied in order to assess their role in the appearance of double polarisation.
7. ACKNOWLEDGEMENT
The research work presented in this paper has been funded by the european commission within the Initial Training Network (ITN) Marie Curie action project BATWOMAN, under the sev-enth framework program (EC grant agreement no. 605867).
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